Her second album One Breath was a more reflective record than her first, an emotional journey through an expansive musical palette.
"I wanted to explore the feeling of being out of control, and how this can be very scary and yet thrilling at the same time," Calvi says. "It's always important to me that the music tells the story as much as the lyrics."
Growing up with her music obsessed Italian father, Anna was exposed to an eclectic array of sounds which ranged from Captain Beefheart to The Stones, to Maria Callas, combined with an early understanding of classical music developed through childhood violin lessons.
She would come to identify the work of 20th century composers, Messiaen, Ravel, and Debussy as an influence, attracted in her words 'to the impressionistic element of the music,' the feel of which she would try to recreate on guitar, an instrument she was compelled to learn on discovering Django Reinhardt and Jimi Hendrix at the age of 13.
Combining all these influences she aimed to make her guitar sound like an orchestra, but only instinctually::
"I don't try and achieve this by using lots of pedals - my Vox amp, a reverb pedal, and then" - she puts her hand somewhere between heart and gut… "it comes from here."
To describe Calvi you'd have to mention PJ Harvey and Siousie Sioux in the same breath .. something she balks at:
"I respect Polly but don't think we've got anything in common. And I have to admit that when I first got likened to Siouxsie I didn't know who she was."
Apart from Nick Cave more of the old guard have come forward with praise.Brian Eno has hailed her as the most exciting female artist since Patti Smith.
What's really fascinating about Calvi is that after studying classical music at university, and after playing guitar in avant garde jazz combos to pay the rent, she only started singing a decade ago.
"I was really fearful of doing it before that point, it was a phobia. Then one day, when I was alone in my parents' house while it was being redecorated, I sang for pretty much the first time. I remember thinking, 'This is quite beautiful.'"
Anna found her musical twin when she met Mally Harpaz in 2006.
"She actually began playing drums with me. But I heard a harmonium being played and just thought it was so beautiful it made me want to cry.
"There's something very timeless and stoic about it. I asked Mally if she'd give it a go.
"She'd never played one before but she's the kind of musician who can just pick stuff up."
The next, and last, recruit to the Calvi less-is-more aesthetic was drummer Daniel Maiden-Wood.
"He's very intuitive. And he listens, which is really rare in a drummer. It wasn't a conscious decision to not have bass when we play live. It was just that I wanted Mally to be the orchestra of the band.
"They are both such great musicians, and I like working with restrictions. I love the rawness of the three of us. And I love space in music.
And it"s exciting, thinking, 'How do I make this sound like a giant string section when I only have this guitar?' That's always the way I've worked."
Coral guitarist Bill Rider-Jones spotted her talent and called up Laurence Bell at Domino and urged him to sign her.
As well as benefiting from Laurence Bell's early support, Anna found yet more outside encouragement from Brian Eno, which came about when a man who happened to be a friend of Eno saw her perform at London's Luminaire, and urged music"s most eminent producer and agent provocateur to check her out.
He did, and was enchanted by her series of wonderful stripped-down performances The Attic Sessions. Eno was so taken by Anna that he asked her out for lunch.
"He was really lovely. I gave him my early basement demos, and he loved them, and since then he's been a real mentor.
"He came along at just the right time. He was the first person from the outside world who heard what I was doing and validated it. It was quite a pivotal moment in my life.
"He sent me a letter saying that the music was full of intelligence, romance and passion, and what more can we want from art? It was like the water at the end of the desert."
The element of light at the end of a long, dark tunnel permeates both the music of, and the story behind, the debut album.
"I just wrote and recorded all the time on an eight-track in my parents" attic. I then spent two-and-a-half years making my first album album in secret in a basement studio.
"It was very unhealthy, actually. I didn't see the sunlight for a long, long time.
"It was such a great experience to then go and work with Rob Ellis."
Rob Ellis is, of course, the producer, composer and musician who has had plenty of experience collaborating with strong female artists through his years with the aforementioned Polly Harvey.
Before her Mercury-nominated album was released, Anna embarked on a one-woman mission to reclaim the ancient and noble idea of the debut single that stands alone from an album she views as a complete work.
Hence the track Jezebel. Anna has toured relentlessly since the turn of the year.
She even completed a support slot on the Grinderman tour – highly appropriate, as band-leader Nick Cave is, of course, the master of love songs that present love and sex as wild and uncontrollable forces. A match made in delicious hell, me thinks!